Section 3.2 (Logarithmic Functions)
In the last section we defined an exponential function f(x) = bx where b is a positive constant other than 1 and
x is any real number. Let’s try to define the inverse of this function.
y = bx (switch x and y)
x = by (solve for y)
=>
=>
???
The inverse function of the exponential function with base b is the logarithmic function with base b
f(x) = log b x is the logarithmic function with base b where y = log b x is equivalent to by = x
There are 2 ways to express the same equation here
Exponent
Logarithmic Form: y = log b x
Exponent
Exponential Form: b y = x
Base
Base
Example: Write the following equations in equivalent exponential form
3 = log 7 x
log 3 9 = y
3 = log b 64
Example: Write the following equations in equivalent logarithmic form
25 = x
1
 2 4
16
b3 = 27
9 3
Given a logarithmic expression, you really ask yourself one question to evaluate the expression…
log 3 9 means 3 to what power gives me 9? => because 32 = 9 then log 3 9 = 2
log 25 5 means 25 to what power gives me 5? => because 25
??
= 5 then log 25 5 = ____
Examples: Evaluate:
log 4 64
log 5
1
125
log 36 6
log 3 5 3
log 9 9
log 8 1
Re-examining the inverse of the exponential function (f(x) = b x), we can now go a little further…
y = bx (switch x and y)
=>
x = by (solve for y)
=>
log b x = y
This gives us the inverse properties of logarithms log b bx = x and b logb x  x
Examples: Evaluate log 7 78 and 3 log 317
Considering that logarithmic functions are the inverses of exponential functions we know that their graphs
should be symmetric about the line y = x…
Example: Graph f(x) = 3x and g(x) = log 3 x in the given coordinate system…
Book problems: 7,11,13,29,37,39,43,47,53,55,63,75,83,97,119
x
f(x) = 3x
-2
-1
0
1
2
x
g(x) = log 3 x
Vertical Asymptote =>
REVIEW CHARACTERISTICS OF GRAPHS (pgs. 404-405)
Table 3.4 on pg. 405 discusses transformations of these logarithmic graphs
Examples: Use these properties to describe the graphs of f(x) log 5 (x + 3) and g(x) = 5 – log 2 x
REVIEW ONLINE HW PROBLEMS 8-10
In looking at the domain of logarithmic functions, consider that in y = log b x that we said b is a positive constant.
So, if a positive number is raised to any power, the sign of the result is ________________.
The domain of a logarithmic function of the form f(x) = log b x is the set of all positive real numbers
Example: Find the domain of f(x) = log 4 (x – 5)
The logarithmic function with base 10 is called the common logarithmic function and the function
f(x) = log 10 x is usually expressed simply as f(x) = log x (the LOG button on calculators refers to the common log
or log 10 x).
Example: Consider some properties of the common log (pg. 407) and use these properties to find log 106
Furthermore, the logarithmic function with base e is called the natural logarithmic function and is usually
expressed as f(x) = ln x (see pgs. 408-410 for properties and note the inverse properties).
Examples: Review online HW problems 13-14.
Book problems: 7,11,13,29,37,39,43,47,53,55,63,75,83,97,119